Question

A piece of ice falls from a height h so that it melts completely. Only one-quarter of the heat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of h is : [Latent heat of ice is $3.4 \times 10^5 \, J/kg$ and $g = 10 \, N/kg$]

• A. 544 km
• B. 136 km
• C. 68 km
• D. 34 k m

Answer 1

The Correct Option is B

To solve the problem, we need to find the height from which a piece of ice falls and melts completely, given that only one-quarter of the heat generated is absorbed by the ice. Here's how to approach it step-by-step:

1. First, let's understand the energy conversion involved:
   • The potential energy of the ice at height h is converted into heat energy when it falls.
   • If m is the mass of the ice, the potential energy U is given by U = mgh.
   • This energy gets converted into heat, and only one-quarter of this heat is absorbed by the ice to melt it.
2. Given that the latent heat of ice (L) is 3.4 \times 10^5 \, \text{J/kg}, the heat required to melt the ice is Q = mL.
3. Only one-quarter of the gravitational potential energy converts into this heat, so we set up the equation: \frac{mgh}{4} = mL.
4. We can simplify the equation:
   \frac{gh}{4} = L
5. Rearranging to find h:
   gh = 4L
   
   
   h = \frac{4L}{g}
6. Substitute the given values (L = 3.4 \times 10^5 \, \text{J/kg}, g = 10 \, \text{N/kg}):
   h = \frac{4 \times 3.4 \times 10^5}{10}
   
   
   h = \frac{1.36 \times 10^6}{10}
   
   
   h = 1.36 \times 10^5 \, \text{m}
7. Convert meters to kilometers:
   h = 136 \, \text{km}

Thus, the correct answer is 136 km. The potential energy lost during the fall is sufficient to melt the ice, considering only a quarter of the generated heat is used for melting. This confirms that the option 136 km is correct.